# A bound for Hall's criterion for nilpotence in semi-abelian categories

## TL;DR

This paper improves the bound in Hall's criterion for nilpotence within semi-abelian categories, establishing the optimal bound and extending Gray's main theorem in this mathematical context.

## Contribution

It provides the best possible bound for Hall's criterion for nilpotence in semi-abelian categories, enhancing Gray's theorem.

## Key findings

- Improved the bound in Hall's criterion for nilpotence
- Proved the bound is optimal
- Extended Gray's main theorem

## Abstract

In this paper, we focus on Hall's criterion for nilpotence in semi-abelian categories, and we improve the bound of Gray's main theorem of [3,Theorem 3.4] (see Main Theorem). And this bound is best possible.